# Harvard Number Theory Seminar, Spring 2019

### Wednesdays 3-4pm at Science Center 507

Date Speaker Title
Feb 6 Rong Zhou (IAS) Motivic cohomology of quaternionic Shimura varieties and level raising
Feb 13 Preston Wake (IAS) Variation of Iwasawa invariants in residually reducible Hida families
Feb 20 Lillian Pierce (Duke) Recent progress on understanding class groups of number fields
Feb 27 Charlotte Chan (Princeton) Affine Deligne--Lusztig varieties at infinite level for GLn
Mar 6 Nicholas Triantafillou (MIT) The method of Chabauty-Coleman-Skolem for restrictions of scalars
Mar 13 Samit Dasgupta (Duke) On Brumer-Stark units
Mar 20 No seminar (Spring Recess)
Mar 27 Kai-Wen Lan (U Minnesota, Twin Cities) Local systems of Shimura varieties: a comparison of two constructions
Apr 4 (Thurs.) Avner Ash (Boston College)
@ Science Center Hall E in the basement
Resolutions of the Steinberg module for GL(n)
Apr 17 Brian Smithling (Johns Hopkins) On Shimura varieties for unitary groups
Apr 24 Yiannis Sakellaridis (Rutgers) A new paradigm for the comparison of trace formulas
May 1 Romyar Sharifi (UCLA) Iwasawa modules in higher codimension

### Organizers

Chi-Yun Hsu and Alison Miller

### Abstracts

Speaker: Rong Zhou (IAS)
Title: Motivic cohomology of quaternionic Shimura varieties and level raising
Abstract: For a smooth scheme of finite type over a field its motivic cohomology groups generalize the usual Chow groups and are an important algebraic invariant. In this talk we will explain how, for the special fibers of certain quaternionic Shimura varieties, its motivic cohomology can encode very rich arithmetic information. More precisely we will show that the cycle class map from motivic cohomology to étale cohomology gives a geometric realization of level raising between Hilbert modular forms. The main ingredient for this construction is a form of Ihara's Lemma for Shimura surfaces which we prove by generalizing a method of Diamond-Taylor.

Speaker: Preston Wake (IAS)
Title: Variation of Iwasawa invariants in residually reducible Hida families
Abstract: We'll discuss a work in progress describing properties of p-adic L-functions of a modular form whose Galois representation is residually reducible. As an application, we prove cases of a conjecture of Greenberg about mu-invariants of Selmer groups of elliptic curves, paying particular attention to the case of X_0(11) with p=5. This is joint work with Rob Pollack.

Speaker: Lillian Pierce (Duke)
Title: Recent progress on understanding class groups of number fields
Abstract: It is natural to think of number fields in families---for example, all number fields of a fixed degree. Correspondingly, we can ask about the distribution of the class number, or of the class group, as the field varies over a family. We will describe a diverse array of recent work that studies for a fixed prime p the size of the p-torsion subgroup of the class group, as the number field varies over a family.

Speaker: Charlotte Chan (Princeton)
Title: Affine Deligne--Lusztig varieties at infinite level for GLn
Abstract: Affine Deligne--Lusztig varieties have been of interest for some time because of their relation to Shimura varieties and the Langlands program. In this talk, we will construct a tower of affine Deligne--Lusztig varieties for GLn and its inner forms. We prove that its limit at infinite level is isomorphic to the semi-infinite Deligne--Lusztig variety of Lusztig and that its cohomology realizes certain cases of automorphic induction and Jacquet--Langlands. This is joint work with A. Ivanov.

Speaker: Nicholas Triantafillou (MIT)
Title: The method of Chabauty-Coleman-Skolem for restrictions of scalars
Abstract: For a number field K and a curve C/K, the Chabauty's method is a powerful p-adic tool for bounding/enumerating the set C(K). The method typically requires that dimension of the Jacobian J of C is greater than the rank of J(K). Since this condition often fails, especially when [K:Q] is large, several techniques have been proposed to augment Chabauty's method. For proper curves, Siksek introduced an analogue of Chabauty's method for the restriction of scalars Res_{K/Q} C that can succeed when the rank of J(O_{K,S}) is as large as [K:Q]*(dim J - 1). Using an analogue of Chabauty's method for restrictions of scalars in the non-proper case, we study the power of this approach together with descent for computing C = P^1 - {0,1,\infty}. As an application, we show that if 3 splits completely in K then there are no solutions to the unit equation x + y = 1 with x,y \in O_K^{\times}$. Speaker: Samit Dasgupta (Duke) Title: On Brumer-Stark units Abstract: In this talk I will discuss: 1) The Brumer-Stark conjecture, which purports the existence of p-units in abelian extensions of totally real fields with valuations at p related to special values of partial zeta functions; 2) The integral Gross-Stark conjecture, which gives a formula for the image of these units under the Artin reciprocity map associated to certain auxiliary extensions; 3) An exact formula for these units developed in my thesis and subsequent work. I will give an update on work in progress with Mahesh Kakde to attack these conjectures. Our key tool is the development of the theory of group-ring families of modular forms. Speaker: Kai-Wen Lan (U Minnesota, Twin Cities) Title: Local systems of Shimura varieties: a comparison of two constructions Abstract: Given a Shimura variety, we can construct two kinds of automorphic local systems, i.e., local systems attached to algebraic representations of certain associated algebraic group. The first one is based on the classical complex analytic construction using double quotients, while the second one is a p-adic analytic construction based on some recently developed p-adic analogue of the Riemann-Hilbert correspondence. I will explain how to compare these two constructions even when the Shimura variety is not of abelian type, and mention some applications. (This is based on joint work with Hansheng Diao, Ruochuan Liu, and Xinwen Zhu.) Speaker: Avner Ash (Boston College) Title: Resolutions of the Steinberg module for GL(n) Abstract: The Steinberg module St(n,K) for the general linear group G of a number field K is the dualizing module for arithmetic subgroups of G. So St(n,K) can be used in studying the cohomology of arithmetic groups, and is especially suitable for computing Hecke operators. I will present several resolutions of St(n,Q) that are helpful in this regard. I will discuss applications to computations for congruence subgroups of SL(4,Z) (with Paul Gunnells and Mark McConnell) and some work related to Serre-type conjectures for GL(n,Z) (with Darrin Doud.) Speaker: Naser Talebizadeh Sardari (U Wisconsin-Madison) Title: The Siegel variance formula for quadratic forms Abstract: We introduce a smooth variance sum associated to a pair of positive definite symmetric integral matrices$A_{m\times m}$and$B_{n\times n}$, where$m\geq n$. By using the oscillator representation, we give a formula for this variance sum in terms of a smooth sum over the square norms of the$B$-th Fourier coefficients of the vector-valued holomorphic Siegel cusp forms which are Hecke eigenforms and obtained by the theta transfer from$O_A$to$Sp_{n}$. By using the Ramanujan bound on the Fourier coefficients of the holomorphic forms when$n=1$, we give a sharp upper bound on this variance. As applications, we generalize the result of Sarnak \cite{Sarnak2}, and give an optimal bound on the diophantine exponent of integral points on any positive definite$d$-dimensional quadric, where$d\geq 3$is odd. This also improves the bound of Ghosh, Gorodnick and Nevo~\cite{GGN} into an optimal bound. Furthermore, we show that almost all even unimodular lattices of dimension$m$represent a given even number greater than$\frac{m}{2\pi e}\$ with respect to Siegel's weights, and this bound is optimal.

Speaker: Brian Smithling (Johns Hopkins)
Title: On Shimura varieties for unitary groups
Abstract: Shimura varieties attached to unitary similitude groups are a well-studied class of PEL Shimura varieties. There are also natural Shimura varieties attached to (honest) unitary groups; these lack a moduli interpretation, but they have other advantages (e.g., they give rise to interesting cycles of the sort that appear in the arithmetic Gan-Gross-Prasad conjecture). I will describe some variant Shimura varieties which enjoy good properties from both of these classes. This is joint work with M. Rapoport and W. Zhang.

Speaker: Yiannis Sakellaridis (Rutgers)
Title: A new paradigm for the comparison of trace formulas
Abstract: Trace formulas are the "mainstream" method for proving Langlands' functoriality, relating the local and automorphic spectra of different groups; in the generalization provided by the relative trace formula, groups can be replaced by appropriate homogeneous spaces. However, the "endoscopic" paradigm of comparisons has nearly reached its limits. In this talk, I will introduce a new paradigm for comparing relative trace formulas, in order to prove instances of (relative) functoriality and relations between periods of automorphic forms and L-functions. More precisely, for an affine spherical variety X=H\G of rank one, I will prove that there is an explicit integral "transfer operator" which transforms the orbital integrals of the relative trace formula for H\G/H to the orbital integrals of the Kuznetsov formula for PGL(2) or SL(2), equipped with suitable non-standard test functions. The operator is determined by the L-value associated to the square of the H-period integral, and the proof uses a deep theory of Friedrich Knop on the cotangent bundles of spherical varieties, viewed as Hamiltonian manifolds. If time permits, I will also discuss other instances of such non-standard comparisons, including Venkatesh's thesis that gave a "beyond endoscopy" proof of functoriality from tori to SL(2), and Hankel transforms which encode the functional equations of L-functions.

Speaker: Romyar Sharifi (UCLA)
Title: Iwasawa modules in higher codimension
Abstract: Classically speaking, Iwasawa theory concerns the growth of p-parts of class groups in towers of number fields of p-power degree. One considers an inverse limit of such groups as a finitely generated, torsion module over a completed pro-p group ring of the tower. Often, this growth can be slow enough that the support of this unramified module lies in codimension two and higher, while invariants like characteristic ideals which one hopes might be described through L-values fail to measure anything beyond codimension one. The typical way around this is to allow ramification at enough primes over p to make the module larger. Yet, the original modules are certainly of arithmetic interest. I'll discuss how, over CM fields, these unramified Iwasawa modules and others satisfy nice reflection principles and how p-adic L-functions tell us something about them. More generally, I'll explain how p-adic L-functions can be used to describe quotients of exterior powers of Iwasawa modules of restricted ramification. This is joint work with F. Bleher, T. Chinburg, R. Greenberg, M. Kakde, and M. Taylor, building on prior joint work of these authors and G. Pappas.